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Fundamental Theorems of Asset Pricing for Good Deal Bounds

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  • Jeremy Staum

Abstract

We prove fundamental theorems of asset pricing for good deal bounds in incomplete markets. These theorems relate arbitrage‐freedom and uniqueness of prices for over‐the‐counter derivatives to existence and uniqueness of a pricing kernel that is consistent with market prices and the acceptance set of good deals. They are proved using duality of convex optimization in locally convex linear topological spaces. The concepts investigated are closely related to convex and coherent risk measures, exact functionals, and coherent lower previsions in the theory of imprecise probabilities.

Suggested Citation

  • Jeremy Staum, 2004. "Fundamental Theorems of Asset Pricing for Good Deal Bounds," Mathematical Finance, Wiley Blackwell, vol. 14(2), pages 141-161, April.
  • Handle: RePEc:bla:mathfi:v:14:y:2004:i:2:p:141-161
    DOI: 10.1111/j.0960-1627.2004.00186.x
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    File URL: https://doi.org/10.1111/j.0960-1627.2004.00186.x
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    Cited by:

    1. Takuji Arai, 2008. "Good deal bounds induced by shortfall risk," Papers 0802.4141, arXiv.org, revised Mar 2010.
    2. Frank Bosserhoff & Mitja Stadje, 2019. "Robustness of Delta hedging in a jump-diffusion model," Papers 1910.08946, arXiv.org.
    3. Balbás, Raquel & Balbás, Alejandro, 2009. "Compatibility between pricing rules and risk measures: the CCVaR," DEE - Working Papers. Business Economics. WB wb090201, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.
    4. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel, 2010. "CAPM and APT-like models with risk measures," Journal of Banking & Finance, Elsevier, vol. 34(6), pages 1166-1174, June.
    5. Dilip Madan, 2015. "Asset pricing theory for two price economies," Annals of Finance, Springer, vol. 11(1), pages 1-35, February.
    6. Soumik Pal, 2006. "Computing strategies for achieving acceptability," Papers math/0607617, arXiv.org.
    7. Vadim Lesnevski & Barry L. Nelson & Jeremy Staum, 2007. "Simulation of Coherent Risk Measures Based on Generalized Scenarios," Management Science, INFORMS, vol. 53(11), pages 1756-1769, November.
    8. Leitner Johannes, 2005. "Optimal portfolios with expected loss constraints and shortfall risk optimal martingale measures," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 49-66, January.
    9. Nicole EL KAROUI & Claudia RAVANELLI, 2008. "Cash Sub-additive Risk Measures and Interest Rate Ambiguity," Swiss Finance Institute Research Paper Series 08-09, Swiss Finance Institute.
    10. Pal, Soumik, 2007. "Computing strategies for achieving acceptability: A Monte Carlo approach," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1587-1605, November.
    11. Bion-Nadal, Jocelyne, 2009. "Bid-ask dynamic pricing in financial markets with transaction costs and liquidity risk," Journal of Mathematical Economics, Elsevier, vol. 45(11), pages 738-750, December.
    12. Marco Frittelli & Giacomo Scandolo, 2006. "Risk Measures And Capital Requirements For Processes," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 589-612, October.
    13. Takuji Arai, 2015. "Good deal bounds with convex constraints," Papers 1506.00396, arXiv.org.
    14. Roorda, Berend & Schumacher, J.M., 2007. "Time consistency conditions for acceptability measures, with an application to Tail Value at Risk," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 209-230, March.
    15. Patrick Cheridito & Michael Kupper & Ludovic Tangpi, 2016. "Duality formulas for robust pricing and hedging in discrete time," Papers 1602.06177, arXiv.org, revised Sep 2017.
    16. PInar, Mustafa Ç. & Salih, AslIhan & CamcI, Ahmet, 2010. "Expected gain-loss pricing and hedging of contingent claims in incomplete markets by linear programming," European Journal of Operational Research, Elsevier, vol. 201(3), pages 770-785, March.
    17. Takuji Arai, 2016. "Good deal bounds with convex constraints: --- examples and proofs ---," Keio-IES Discussion Paper Series 2016-017, Institute for Economics Studies, Keio University.
    18. Hirbod Assa & Nikolay Gospodinov, 2018. "Market consistent valuations with financial imperfection," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(1), pages 65-90, May.
    19. Takuji Arai & Masaaki Fukasawa, 2011. "Convex risk measures for good deal bounds," Papers 1108.1273, arXiv.org.
    20. Balbás, Alejandro & Balbás, Raquel & Garrido, José, 2010. "Extending pricing rules with general risk functions," European Journal of Operational Research, Elsevier, vol. 201(1), pages 23-33, February.
    21. Steven Kou & Xianhua Peng & Chris C. Heyde, 2013. "External Risk Measures and Basel Accords," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 393-417, August.
    22. Takuji Arai, 2017. "Good Deal Bounds With Convex Constraints," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-15, March.
    23. Balbás, Alejandro, 2008. "Capital requirements: Are they the best solution?," DEE - Working Papers. Business Economics. WB wb087114, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.

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